The invention relates to a method for analyzing optical intensity distributions of objects in an optical projection system, and more particularly to a method for analyzing optical intensity distributions for imaging simulation of mask patterns used in semiconductor photo-lithography processes.
It has been well known in the field of semiconductors that the optical projection system is useful for photo-lithography needed to prepare any desired resist patterns to be used in fabrication processes of semiconductor devices such as semiconductor integrated circuits. FIG. 1 illustrates a configuration of the optical projection system has a light source 1 emitting a ray, first to third condenser lenses 2, 4 and 6, a mask 3 including a pattern 101 and a pupil 5 as well as an image screen 7. The above elements comprising the optical projection system are aligned on a center axis. The first condenser lens 2 is placed to face the light source 1 for receiving a spreading ray emitted from the light source 1 and subsequent conversion of the spreading ray into a parallel ray. The second condenser lens 4 is placed between the first and third condenser lenses 2 and 6. The mask 3 including the pattern 101 is placed between the first and second condenser lenses 2 and 4 so that the parallel ray is irradiated on the mask 3. The parallel ray is then transmitted through the mask 3 with the pattern 101 to the second condenser lens 4 by which the parallel ray is converted into a unifocal ray. The unifocal ray has a focus point at an intermediate point of a half-distance of a distance between the second and third condenser lenses 4 and 6. At the intermediate point between the second and third condenser lenses 4 and 6, the pupil 5 is positioned. The pupil 5 has a semicircular aperture through which the ray is transmitted and a semicircular opaque counterpart for restricting the transmission of the ray. The transmitted unifocal ray is received by the third condenser lens 6 for conversion thereof into the parellel ray that is subsequently irradiated on the imaging screen 7 thereby the image of the pattern 101 is projected on the image screen 7.
In the above optical projecting system, the pupil 5 having the semicircular aperture and the opaque semicircular counterpart may generate a ray amplitude distribution or a ray intensity distribution that may be expressed by a parallel-displaced Fourier transformation that is subjected to a parallel displacement of an original Fourier transformation of the pattern 101. The original Fourier transformation is given by F(f, g). An intensity distribution on the pupil 5 of a ray that has been emitted from a point (p.sub.i, q.sub.i) on the light source 1 is then given by a parallel-displaced Fourier transformation F(f-p.sub.i /.lambda., g-q.sub.i /.lambda.). A relationship of the transmission and cur off of the ray may be expressed by a pupil function P(-.lambda.Zf, -.lambda.Zg) in which transparent and opaque portions are expressed by "1" and "0" respectively. Further, it is required to consider an influence of aberration by the lenses, which may be expressed by an exponential function exp(j(2.pi./.lambda.)W(-.lambda.Zf, -.lambda.Zg)). The product of the parallel-displaced Fourier function and the influence of the aberration is defined as the following function K(f, g). EQU K(f, g)=P(-.lambda.Zf, -.lambda.Zg)exp(j(2.pi./.lambda.)W(-.lambda.Zf, -.lambda.Zg)) (1)
where Z is the distance between pupil 5 and the image screen 7. The ray transmitted through the pupil 5 has such an intensity distribution or an amplitude distribution as given by F(f-p.sub.i /.lambda., g-q.sub.i /.lambda.)K(f, g). When the ray is propagated through the third condenser lens 6 onto the image screen 7, then the ray has an intensity distribution or an amplitude distribution that corresponds to one subjected to Fourier transformation of the intensity distribution of the ray at the pupil 5. Accordingly, the intensity distribution on the image screen 7 is given by the following equation. ##EQU1##
A calculation method of the equation (2) is disclosed in Proceedings of Kodak Microelectronics Seminar INTERFACE'85. pp. 115-126, "MODELING AERIAL IMAGES IN TWO AND THREE DIMENSIONS". This reference describes about methods of creating two-dimensional imaging models by use of the projection system with computer added design model analysis as well as creating three dimensional of a ray propagating in directions oblique to a surface.
A conventional light intensity distribution analyzing method is to be carried out by the following steps P1 to P6. FIG. 2 is a flow chart of the conventional light intensity distribution analyzing method. In a step P1, a mask pattern is converted into a bit map. A calculation of of Fourier transformation F(f, g) of the mask is carried out in a step P2. The aberration function k(f, g) is calculated based on the pupil equation (1) in a step P3. In a step P4, the Fourier transformation F(f, g) is parallel-displaced by a point (P.sub.i, q.sub.i) on the light source 1 to obtain the parallel-displaced Fourier transformation F(f-q.sub.i /.lambda., g-q.sub.i /.lambda.) for subsequent calculation of the product of the parallel-displaced Fourier transformation F(f-q.sub.i /.lambda., g-q.sub.i /.lambda.) and the aberration function K(f, g) to find the equation F(f-q.sub.i /.lambda., g-q.sub.i /.lambda.)K(f, g). In a step P5, an inverse Fourier transformation of the equation F(f-q.sub.i /.lambda., g-q.sub.i /.lambda.)K(f, g) is carried out by fast Fourier transformation to obtain T(F, K). In a step 6, the steps P4 and P5 are repeated for all the elements on the light source 1 for subsequent calculation of a sum of squares of absolute values of individual T(F, K) or .SIGMA..vertline.T(F, K).vertline..sup.2.
The mask pattern 101 subjected to the Fourier transformation may be a complicated pattern. According to the above conventional method, it is required to calculate the Fourier transformation of any mask patterns including complicated patterns. Methods for calculating the Fourier transformation of any mask patterns including complicated patterns are disclosed in IEEE Transactions on Electron Devices, Vol. ED-31, No. 6, June 1984, "The Phase-Shifting Mask II: Imaging Simulations and Submicrometer Resist Exposures".
In the step 2, the Fourier transformation of the mask pattern may be calculated by the fast Fourier transformation. As well known, the fast Fourier transformation is to be carried out by using redundancy of determinations to calculate descrete Fourier transformation in which a function is sampled at a predetermined distance to obtain discrete values. The fast Fourier transformation is able to convert any patterns. Nevertheless, a result of the calculation of the discrete values would not strictly correspond to a result of the normal Fourier transformation. FIGS. 3A and 3B illustrate waveforms as results of the normal and fast Fourier transformations respectively. The waveforms obtained by the fast Fourier transformation are different from the waveform of the normal Fourier transformation. The mask pattern 101 may be expressed by a step function in which a transparent portion and an opaque portion are set as "1" and "0" respectively. A Fourier-transformed step function has a frequency component range extending to a high frequency region. In sampling processes, the Fourier transformation receives such a frequency restriction as eliminating a high frequency component. As a result, the obtained waveforms have base portions being overlapped on base portions of the adjacent waveforms as illustrated in FIG. 3B. For that reason, the calculation result of the fast Fourier transformation is necessarily different from the result of the normal Fourier transformation. This means that using the fast Fourier necessarily leads to the lowering of a calculation accuracy for calculating the mask pattern. The accuracy of the calculation by the fast Fourier transformation depends upon the number of mashes for sampling in the fast Fourier transformation. A large number of the meshes permit increasing a period of the waveforms illustrated in FIG. 3B. This leads to overlap-free waveforms thereby the calculation accuracy is improved. However, a large number of the meshes requires a time consuming calculation process of the invert Fourier transformation by the fast Fourier transformation in the step P5. Using N.times.N meshes requires a time proportional to Nlog.sub.2 N.times.Nlog.sub.2 N. The increase of the mesh number N requires a considerable increase of the time for the fast Fourier transformation. As described above, the step P5 for the fast Fourier transformation is repeated for all the elements on the light source 1 in the step P6. Accordingly, the step P6 requires a considerable long time.
Consequently, the above conventional intensity distribution analyzing method is engaged with the disadvantages in poor calculation accuracy caused by the frequency restriction in sampling processes of the fast Fourier transformation. The above conventional intensity distribution analyzing method is also engaged with the disadvantages in requirement in the time consuming calculation process of the invert Fourier transformation by the fast Fourier transformation.